Can interacting quantum field theory describe more than just scattering? From my understanding we do not yet know how to make much out of interacting QFT other than scattering amplitude at asymptotic infinity. (Correct me if I misunderstand.) But path integral, in principle should allow us to compute different sorts of transition probabilities. So is lack of understanding more about constructing observables such that we can test with measurements? That is, is this lack of understanding about our current lack of how to interpret states at times not sent to asymptotic infinity?
 A: In principle, we can treat interacting QFT just like any other quantum theory, in terms of time-evolving states in the Schrödinger picture (for example), without relying on the asymptotic past/future.
Conceptually, a QFT is a net of algebras of local observables
(as explained in Haag's book Local Quantum Physics),
and this can be realized explicitly in many interacting QFTs
if we don't mind treating spacetime as a discrete lattice
(as explained in Montvay and Munster's book Quantum Fields on a Lattice). In those cases where we know how to construct the model on a discrete spatial lattice (which excludes chiral non-abelian gauge theories like the Standard Model, but includes both QED and QCD), we can even write down the Schrodinger equation for a general state-vector like this:
$$
  i\frac{\partial}{\partial t}\Psi[t,\phi]
 = H\Psi[t,\phi]
\tag{1}
$$
where $\Psi[t,\phi]$ is the wavefunctional
as a function of time $t$ and all of the field variables $\phi$, which may include scalar fields, spinor fields, and
gauge fields. The Hamiltonian $H$ is expressed in terms of field operators, which are in turn expressed in terms of things like derivatives with respect to the field variables $\phi$.  So, in principle, we can treat interacting QFT just like any other quantum theory, in terms of time-evolving states, without relying on the asymptotic past/future.
However, relating
those local observables to specific physically-recognizable things like
electrons and protons is difficult.
The usual approach of looking for poles in time-ordered functions is fine for scattering experiments,
but when it comes to studying the time-evolution of state-vectors
with clear physical interpretations, we're pretty much stuck
— we don't even have an explicit expression for the vacuum state
in most interacting QFTs, much less states
with specified configurations of particles.
Such states exist mathematically (modulo the usual ambiguities
in what we mean by "particle"), but actually constructing
them in interesting models
 seems to be beyond our current mathematical abilities.
I think this is the single biggest obstacle to teaching QFT
in a really satisfying way.
The meaning of renormalization is a solved problem
("What is renormalization?", https://arxiv.org/abs/hep-ph/0506330),
but conceptually-simple things like writing down an
explicit representation for a single-electron state
are still beyond our reach, as far as I know.
As an example of what we do and don't know how to do, you might be interested in this classic paper by Feynman, where he studies the Schrodinger equation in SU(2) gauge theory in (continuous) three-dimensional spacetime:


*

*Feynman (1981), "The qualitative behavior of Yang-Mills theory in 2 + 1 dimensions", Nuclear Physics B 188: 479-512,
https://www.sciencedirect.com/science/article/pii/0550321381900055
(The Schrodinger equation is written in "solved" form in equation (30) on page 498, using the path-integral trick that helped make Feynman famous.)
